MATHEMATICAL MODEL FOR THERMAL CRACKING
OF HYDROCARBONS
ETHANE CRACKING
The simulation of a thermal cracking coil requires integration of a
set of
• Mass balance
• Energy balance
• Momentum balance equations.
Mass Balance
dFj
πdt2
---- = - Σ(sijri) ----dz
4
(1)
Reaction Mechanism for Ethane Cracking
C2H6
→ CH3* + CH3*
CH3* + C2H6 → CH4 + C2H5*
C2H5*
→ C2H4 + H*
H* + C2H6
→ H2 + C2H5*
H* + H*
→ H2
H* + CH3*
→ CH4
H* + C2H5*
→ C2H6
C2H5* + CH3* → C3H8
C2H5* + C2H 5*→ C4H10
Reaction scheme for ethane cracker model
E (kcal/kmol) A
↔ C2H4 + H2
C2H6
C2H4 + 2H2
2.50E08
72712
1.24E15
↔ 2CH4
58600
C2H4 → 0.125C4H8 + 0.125C4H10 + 0.25 C4H6 + 0.125H2
C2H4
1.80E09
→
1/3 C6H6 + H2
66800
C2H4
4.00E08
→
C2H2 + H2
61600
C2H4
→ 2C + 2H2
49000
C2H4 + C2H6 ↔ 0.381C3H8 + 0.952C3H6 + 0.62H2
92400
0.94E14
5.81E05
Reaction rates
r1 = A1 exp(-E1/RT) (pp(C2H6) - pp(C2H4)*pp(H2)/Kp1)
r2 = A2exp(-E2/RT) pp(C2H4) √pp(C2H6)pp(H2) pp(CH4)/Kp2)
r3 = 0.012 r1 P
r4 = A4 exp(-E4/RT) pp(C2H4)2
r5 = A5 exp(-E5/RT) pp(C2H4)2
r6 = A6 exp(-E6/RT) pp(C2H4)2
r7 = A7 exp(-E7/RT) (pp(C2H4) - pp(C2H4)*pp(H2)/Kp1)
Material balance equations
dC2H6 /dZ = Πd2/4 (-r1-r7)
dCH4 /dZ = Πd2/4 (2 r2)
dC2H4 /dZ = Πd2/4 (r1 - r2 - r3 - r4 - r5 - r6 - r7)
dC3H8 /dZ = Πd2/4 (0.381r7)
dC3H6/dZ = Πd2/4 (0.952r7)
dC2H2 /dZ = Πd2/4 (r5)
dH2 /dZ = Πd2/4 (r1 - 2r2
+ 0.125 r3
dC4H10 /dZ = Πd2/4 (0.125r3)
dC4H8 /dZ = Πd2/4 (0.125r3)
dC4H6 /dZ = Πd2/4 (0.25r3)
dC6H6 /dZ = Πd2/4 (0.333r4)
dC /dZ = Πd2/4 (2r6)
+ r4 + r5 + 2r6 + 0.62r7)
Energy Balance
dT
 =
dz
1
πdt2
 [Q(z) πdt +  ri (-∆Hi )]
∑FjCpj
4
Cp = specific heat
Q
= heat flux
dt = coil diameter
ri = rate of reaction
∆H = heat of reaction
• In order to avoid the complications of solving the above energy
balance equation with the heat transfer coefficients, specific
heats of each component, the heat flux profiles and heat of
reaction, we have applied directly temperature profiles being
used in industrial ethane cracker across the length of the reactor,
in a polynomial form.
Momentum balance
• The pressure drop equation along the length of the cracking coil
was derived by rapid estimates.
• In most empty tubular reactors kinetic energy changes are
negligible and only the friction losses need be considered.
• The friction losses can be obtained from
∆P
f = 
ρf z u2/2 gc
• In the Reynolds number ranges of steam cracking flow rates the
friction factor for smooth tubes can be calculated using
f = 0.184/NRe0.2
where NRe = DG/µm .
Pressure drop was calculated by
combining the above two Equations.
1.77 × 10-7 G1.8 µm 0.2 z
∆P = -------------------------------D1.2 ρf
∑ yi µi √Mi
µm = 
∑ yi √Mi
The viscosity of individual component is calculated by
µi =
33.3 (√ MTc) [f(1.33Tr)]

Vc0.66
f(1.33Tr) = 1.058 Tr
0.645
-
0.261

(1.9Tr) 0.8log(1.9Tr)
• The above set of continuity equations for each species along
with energy and pressure drop equations are numerically
integrated using fourth order Runge-Kutta method to obtain the
axial profiles of conversion, temperature, and pressure.
COKING MODEL
• Thermal cracking of hydrocarbons is always accompanied with
the formation of coke which deposited on the walls of the coil.
• Coke may be formed either directly from the feed stock and/or
from the products.
Many components from the feed and
products are capable of yielding coke which are called the coke
precursors.
• The coke deposited in the coil and in the TLX hampers heat
transfer and thereby requiring higher tube skin temperature.
• The coke deposition also reduces coil diameter which in turn
leads to higher inlet pressures which are detrimental to ethylene
yield.
• The temperature increase of the tube wall and pressure drop
necessitate shutdown of the plant for decoking.
• Rate of coke deposition depends on several factors such as feed
stock, operating conditions, pyrolysis coil design, its material of
construction and pre treatments given to the inner walls of the
coil.
Coking kinetics and rate of coke deposition along
the length of the cracking coil as a function of time
have to be incorporated in the main pyrolysis model
to able to simulate run length.
•
This helps in predicting the coke thickness inside
the coil which in turn predicts the run length of
cracking coil for a given set of operating parameters
and a desired yield pattern.
•
The present model considers ethane, and ethylene
as potential coke precursors for run length
simulation of ethane cracking.
Coking reaction scheme for ethane cracker model
E, Kcal/gmol A
Ethane → coke
Ethylene → coke
76.9
49.61
Reaction
order (n)
2.01E15
1.83E10
1
1
The rate of coke formation can be expressed as
m
rc = ∑ rci m is number of
precursors
i=1
rci = Ai exp(-Ei / RTf) ci ni
where ci is the concentration of the coke precursor
which can be expressed in terms of partial pressure
and temperature.
• The initial gas temperature profile was maintained constant for
the complete run length.
• The concentrations of the precursors, Ci, are generated by main
reaction model along the length of coil.
The average of
concentrations at the entrance and exit of each pass is taken as
the concentration of that particular pass.
• The continuity equation for coking is integrated by incrementing
time in stepwise.
• We have taken 24 h as step length.
• The thickness of the coke deposited, bck , is calculated using the
following relation (Lichtenstein, 1964).
di
αck rc ts
bck =  ( 1 - exp(  ))
2
2ρck
The pressure drop in coked tube is calculated using
∆Pck = ∆P(Gck/G)1.8 (d/dck)1.2( ρck/ρ)
where ∆P is clean tube pressure drop.
• The total increase in inlet pressure is calculated and checked
with the limiting value. Once the increase in the inlet pressure
exceeds the limiting value the calculations are stopped and the
corresponding time is reported as run length.
Decoking is considered necessary when one of
the following criteria is satisfied
1. Inlet pressure exceeding the limiting value
2. External tube skin temperature exceeding
1080°C
External tube skin temperature
• The external tube skin temperature is calculated by using the
following relations (Rase, 1977).
Tw = T + ∆Tf + ∆Tck + ∆Tw
Q0do
Q0do
Q0 do bw
∆Tf =  ; ∆Tck =  ; ∆Tw = 
hi dck
λckdck
λw d
• where T is fluid temperature and ∆Tf is temperature drop across
the film, ∆Tck is temperature drop across the coke and ∆Tw is
temperature drop across the tube wall.
• hi, inside heat transfer coefficient is calculated using Dittus
Boelter relation
λf dck G 0.8 Cp µ
hi = 0.023 − (  ) (  )
dck
µ
λf
THE INPUT
• Molecular weights
• Critical properties
• Step size for calculations
• Temperature profile equations
• Coil geometry
• Kinetic parameters
0.4
• Feed rate (Flow rate of ethane per coil, t/h)
• Dilution ratio
• Crossover temperature
• Coil outlet temperature
• Coil inlet pressure
THE OUTPUT
∗ Concentration profile
∗ Temperature profile
∗ Pressure profile
∗ Product yields and Run length with varying feed stock quality
and operating conditions
∗ External tube skin temperature
PROPANE CRACKING
Reactions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
C3H8
→ C2H4 + CH4
C3H8
↔ C3H6 + H2
C3H8
→ 0.5C4H10 + 0.5C2H6
C3H8
→ 0.5CH4 + 0.5C3H6 + 0.5C2H6
C2H6
→ C2H4 + H2
C2H6
→ CH4 + 0.5C2H4
C2H6
→ 0.5CH4 + 0.5C3H8
C3H6
→ 1.5C2H4
C3H6 + H2 → CH4 + C2H4
C3H6
→ C2H2 + CH4
C2H4 + H2 → C2H6
C2H4
→ C2H2 + H2
C2H4
→ 0.667C3H6
2C2H2 + H2 → C4H6
C3H6 + C2H2 → C5s
Reaction rates
(kmol/m3 sec)
r1 =
r2 =
r3 =
r4 =
r5 =
r6 =
r7 =
r8 =
r9 =
r10 =
r11 =
r12 =
r13 =
r14 =
r15 =
A(sec-1)
E
or * (m3/kmol s) (kcal/kmol)
kp1 pp(C3H8)
2.62E09
kp2 pp(C3H8)
2.00E09
kp3 pp(C3H8)
2.20E09
kp4 pp(C3H8)
1.10E09
kp5 pp(C2H6)
0.34E13
kp6 pp(C2H6)
3.90E12
kp7 pp(C2H6)
0.20E11
kp8 pp(C3H6)
0.99E10
kp9 pp(C3H6)pp(H2)*
1.00E15
kp10 pp(C3H6)
1.40E10
kp11 pp(C2H4)pp(H2)* 0.68E13
kp12 pp(C2H4)
7.70E13
kp13 pp(C2H4)
1.40E11
kp14 pp(C2H2)pp(H2)* 9.90E10
kp15 pp(C2H2)pp(C3H6)* 9.00E14
44000
44000
54000
48000
60000
67000
59000
44200
60000
50000
52000
76000
51000
36000
51000
Material Balance
d(CH4)/dz = r1 + 0.5r4 + r6 + 0.5r7 + r9 + r10
d(C2H4)/dz = r1 + r5 + 0.5r6 + 1.5r8 + r9 - r11 - r12 - r13
d(C2H6)/dz = 0.5r3 + 0.5r4 - r5 - r6 - r7 + r11
d(C3H8)/dz = -r1 - r2 - r3 - r4 + 0.5r7
d(C3H6)/dz = r2 + 0.5r4 - r8 - r9 - r10 + 0.667r13 - r15
d(C2H2)/dz = r10 + r12 - 2r14 - r15
d(H2)/dz = r2 + r5 + r12 - r9 - r11 - r14
d(C4H10)/dz = 0.5r3
d(C4H6)/dz = r14
d(C5s)/dz = r15
REACTION SCHEME FOR LPG CRACKING
No. Reaction
Source
A
E, Kcal/gmol
1. C2H6
↔ C2H4 + H2
3.052E13
64.13
E
2. 2C2H6
→ C3H8 + CH4
3.750E12
65.25
E
60.43
E
50.60
P
51.29
P
3. C2H4 + C2H6 → C3H6 + CH4
4. C3H8
→ C2H4 + CH4
5. C3H8
↔ C3H6 + H2
6. C3H8 + C2H4 → C2H6 + C3H6
P
7. C3H6
8. C2H6
P
9. 2C3H6
↔ C2H2 + CH4
↔ C2H4 + H2
→ 3C2H4
6.083E13
4.992E10
6.888E10
7.036E13
4.404E11
3.652E13
1.544E11
59.06
59.39
65.21
55.80
10. C2H2 + C2H4 → C4H6
P
0.160E12
41.26
11. C3H6 + C2H6 → 1-C4H8 + CH4
P
2.500E14
60.01
12. 2C3H6
P
→ 0.5C6+ + 3CH4
0.123E08
P
45.50
P
13. n-C4H10
→ C3H6 + CH4
14. n-C4H10
NB
→ 2C2H4 + H2
7.500E14
70.68
15. n-C4H10
NB
→ C2H4 + C2H6
4.099E12
61.31
16. n-C4H10
NB
↔1-C4H8 + H2
9.637E12
62.36
17. C3H6 + H2
NB
→ C2H4 + CH4
4.770E09
35.0
18. C2H2 + C2H4 → C4H6
NB
1.000E13
59.64
0.0920E12
NB
41.26
19. i-C4H10
IB
↔ i-C4H8 + H2
9.046E11
54.40
20. i-C4H10
IB
→ C3H6 + CH4
5.000E11
54.43
21. i-C4H10 + C2H4 → 2-C4H8 + C2H6 1.320E09
30.24
IB
22. i-C4H10
→
C3H4 + CH4
2.954E15
74.0
IB
23. C3H4
→
C6+
3.504E04
14.5
IB
24. C2H2 + C2H4 → C4H6
IB
0.0920E12
41.26
THERMAL CRACKING OF ETHANE-PROPANE MIXTURES
Reaction Scheme for the Cracking of Mixtures of Ethane and
Propane
∗ The combination of both ethane and propane cracking models
enabled a molecular reaction scheme for the cracking of
mixtures of both the components.